Theorem isoml 37252

Description: The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)

Hypotheses

Ref	Expression
isoml.b	⊢ 𝐵 = (Base‘𝐾)
isoml.l	⊢ ≤ = (le‘𝐾)
isoml.j	⊢ ∨ = (join‘𝐾)
isoml.m	⊢ ∧ = (meet‘𝐾)
isoml.o	⊢ ⊥ = (oc‘𝐾)

Assertion

Ref	Expression
isoml	⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦

Allowed substitution hints:   ∨ (𝑥,𝑦)   ≤ (𝑥,𝑦)   ∧ (𝑥,𝑦)   ⊥ (𝑥,𝑦)

Proof of Theorem isoml

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6774	. . . 4 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2		isoml.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2796	. . 3 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4		fveq2 6774	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5		isoml.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
6	4, 5	eqtr4di 2796	. . . . . 6 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
7	6	breqd 5085	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦 ↔ 𝑥 ≤ 𝑦))
8		fveq2 6774	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9		isoml.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
10	8, 9	eqtr4di 2796	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
11		eqidd 2739	. . . . . . 7 ⊢ (𝑘 = 𝐾 → 𝑥 = 𝑥)
12		fveq2 6774	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
13		isoml.m	. . . . . . . . 9 ⊢ ∧ = (meet‘𝐾)
14	12, 13	eqtr4di 2796	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = ∧ )
15		eqidd 2739	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → 𝑦 = 𝑦)
16		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
17		isoml.o	. . . . . . . . . 10 ⊢ ⊥ = (oc‘𝐾)
18	16, 17	eqtr4di 2796	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (oc‘𝑘) = ⊥ )
19	18	fveq1d 6776	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑥) = ( ⊥ ‘𝑥))
20	14, 15, 19	oveq123d 7296	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)) = (𝑦 ∧ ( ⊥ ‘𝑥)))
21	10, 11, 20	oveq123d 7296	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))
22	21	eqeq2d 2749	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) ↔ 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))))
23	7, 22	imbi12d 345	. . . 4 ⊢ (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
24	3, 23	raleqbidv 3336	. . 3 ⊢ (𝑘 = 𝐾 → (∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
25	3, 24	raleqbidv 3336	. 2 ⊢ (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
26		df-oml 37193	. 2 ⊢ OML = {𝑘 ∈ OL ∣ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))))}
27	25, 26	elrab2 3627	1 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  lecple 16969  occoc 16970  joincjn 18029  meetcmee 18030  OLcol 37188  OMLcoml 37189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-ov 7278  df-oml 37193

This theorem is referenced by:  isomliN  37253  omlol  37254  omllaw  37257